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Maths · Similarity and congruence

More similarity

Use similar triangles formed by parallel lines, and use the fact that areas of similar shapes scale by the square of the length scale factor.

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  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    What is the scale factor from 6 to 9?

    Show answerHide answer

    1.5

  2. 2

    Work out \(1.5^2\).

    Show answerHide answer

    2.25

  3. 3

    What do corresponding angles on parallel lines do?

    Show answerHide answer

    They are equal

  4. 4

    Work out \(\sqrt{\dfrac{27}{12}}\).

    Show answerHide answer

    1.5

  5. 5

    If \(k = 3\), what is \(k^2\)?

    Show answerHide answer

    9

Learning Objectives

  1. 1Find similar triangles created by parallel lines.
  2. 2Find missing lengths using scale factors.
  3. 3Use the area scale factor \(k^2\).
  4. 4Work backwards from an area ratio to a length ratio.

PARALLEL LINES

If DE is parallel to BC, then triangle ADE is similar to triangle ABC.

The corresponding angles are equal, and the angle at A is common.

Finding a Length

In triangle ABC, DE is parallel to BC. \(AD = 4\) cm, \(DB = 2\) cm and \(DE = 5\) cm. Find \(BC\).

Show the solutionHide the solution
  1. 1 \(AB = AD + DB\) \(4 + 2 = 6\)
  2. 2 Scale factor from ADE to ABC \(\dfrac{6}{4} = 1.5\)
  3. 3 \(BC = DE \times 1.5\) \(5 \times 1.5\)

Answer\(BC = 7.5\) cm

Watch the Sides

The whole side, not just the piece, is the corresponding length.

  • Use AB, not DB

    The similar triangles are ADE and ABC, so AD corresponds to AB.

  • Add the parts

    \(AB = AD + DB\).

  • Label the two triangles

    Write them in matching order: ADE and ABC.

AREA SCALE FACTOR

If two shapes are similar with length scale factor \(k\), their areas are in the ratio \(1 : k^2\).

Doubling all lengths multiplies the area by 4; tripling multiplies it by 9.

Finding an Area

Two similar rectangles have lengths 12 cm and 18 cm. The area of the smaller is 40 cm². Find the area of the larger.

Show the solutionHide the solution
  1. 1 Length scale factor \(\dfrac{18}{12} = 1.5\)
  2. 2 Area scale factor \(1.5^2 = 2.25\)
  3. 3 Area of the larger \(40 \times 2.25\)

Answer90 cm²

From Areas to Lengths

Two similar triangles have areas 12 cm² and 27 cm². Find the ratio of their lengths in its simplest form.

Show the solutionHide the solution
  1. 1 Ratio of areas \(12 : 27 = 4 : 9\)
  2. 2 Take the square root \(\sqrt{4} : \sqrt{9} = 2 : 3\)

AnswerThe ratio of the lengths is \(2 : 3\).

Scale Factors

For similar shapes with length scale factor \(k\).

  • Lengths

    Scale factor: \(k\). Example, \(k = 3\): 3

  • Areas

    Scale factor: \(k^2\). Example, \(k = 3\): 9

  • Volumes

    Scale factor: \(k^3\). Example, \(k = 3\): 27

Shadows and Similar Triangles

A 1.5 m post casts a shadow of 2 m at the same time as a tree casts a shadow of 12 m. Draw two similar right-angled triangles and find the height of the tree. If the post's triangle has area 1.5 m², find the area of the tree's triangle.

1. Draw both triangles.

2. Find the length scale factor.

3. Square it for area.

A good answer shows: Scale factor \(12 \div 2 = 6\), so the tree is \(1.5 \times 6 = 9\) m tall. The area scale factor is \(6^2 = 36\), so the area is \(1.5 \times 36 = 54\) m².

Can I...?

  1. 1Spot similar triangles from parallel lines.
  2. 2Use the whole side, not part of it.
  3. 3Find a missing length.
  4. 4Use \(k^2\) for areas.
  5. 5Find an area from a length ratio.
  6. 6Find a length ratio from an area ratio.
  7. 7Write the ratio in simplest form.
  8. 8Explain my reasoning.

Summary & Exam Focus

  • A line parallel to a side makes a similar triangle.
  • Match corresponding sides in the same order.
  • Area scale factor \(= k^2\).
  • Take a square root to go from area ratio to length ratio.

Exam focus

In triangle ABC, DE is parallel to BC. \(AD = 4\) cm, \(DB = 2\) cm and \(DE = 5\) cm. Work out the length of BC. (3 marks) (3 marks)

The similar triangles are ADE and ABC. Use AB \(=\) AD \(+\) DB in the scale factor, not DB alone.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Similar triangles
Triangles with equal angles and proportional sides.
Scale factor
The multiplier for lengths.
Area scale factor
The square of the length scale factor.
Parallel
Lines that never meet.
Corresponding angles
Equal angles in matching positions on parallel lines.
Ratio
A comparison of two quantities.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Work out 3 marks

    In triangle ABC, DE is parallel to BC. \(AD = 4\) cm, \(DB = 2\) cm and \(DE = 5\) cm. Work out the length of BC.

    A triangle ABC with a parallel line DE, AD = 4 cm, DB = 2 cm and DE = 5 cm, with BC to be found.
    Show answerHide answer

    Model answer

    Triangles ADE and ABC are similar. \(AB = 6\), so the scale factor is \(\dfrac{6}{4} = 1.5\). \(BC = 5 \times 1.5 = 7.5\) cm.

    Mark scheme

    • Similar triangles ADE and ABC — M1
    • Scale factor \(\dfrac{6}{4}\) — M1
    • 7.5 — A1
  2. Question 2 Work out 3 marks

    Two similar rectangles have lengths of 12 cm and 18 cm. The area of the smaller rectangle is 40 cm². Work out the area of the larger rectangle.

    Show answerHide answer

    Model answer

    The length scale factor is \(\dfrac{18}{12} = 1.5\), so the area scale factor is \(1.5^2 = 2.25\). The area is \(40 \times 2.25 = 90\) cm².

    Mark scheme

    • Length scale factor 1.5 — M1
    • \(1.5^2 = 2.25\) — M1
    • 90 — A1
  3. Question 3 Work out 3 marks

    Two similar triangles have areas of 12 cm² and 27 cm². Work out the ratio of the lengths of their corresponding sides. Give your answer in the form \(1 : n\) or in its simplest form.

    Show answerHide answer

    Model answer

    The area ratio is \(12 : 27 = 4 : 9\). The length ratio is \(\sqrt{4} : \sqrt{9} = 2 : 3\).

    Mark scheme

    • \(4 : 9\) — M1
    • Square roots — M1
    • \(2 : 3\) — A1
  4. Question 4 Work out 3 marks

    Two similar shapes have corresponding lengths in the ratio \(3 : 5\). The area of the smaller shape is 36 cm². Work out the area of the larger shape.

    Show answerHide answer

    Model answer

    The area ratio is \(3^2 : 5^2 = 9 : 25\). The area is \(36 \times \dfrac{25}{9} = 100\) cm².

    Mark scheme

    • \(9 : 25\) — M1
    • \(36 \times \dfrac{25}{9}\) — M1
    • 100 — A1
  5. Question 5 Work out 3 marks

    A 1.5 m post casts a shadow 2 m long at the same time as a tree casts a shadow 12 m long. Work out the height of the tree.

    Show answerHide answer

    Model answer

    The triangles are similar. The scale factor is \(12 \div 2 = 6\). The tree is \(1.5 \times 6 = 9\) m tall.

    Mark scheme

    • Scale factor 6 — M1
    • \(1.5 \times 6\) — M1
    • 9 — A1
  6. Question 6 Explain 3 marks

    Explain why triangle ADE is similar to triangle ABC when DE is parallel to BC.

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    Model answer

    Angle \(ADE\) = angle \(ABC\) and angle \(AED\) = angle \(ACB\) (corresponding angles on parallel lines). Angle \(A\) is common. All three angles are equal, so the triangles are similar.

    Mark scheme

    • Corresponding angles on parallel lines — M1
    • Angle A common — M1
    • All angles equal, so similar — C1

Quick check

  1. DE is parallel to BC in triangle ABC. Which triangle is similar to ABC?

    1. AADE
    2. BDBC
    3. CBDE
    4. DCDE
    Show answerHide answer

    A: ADE

    The small triangle ADE.

  2. Two similar shapes have length scale factor 4. What is the area scale factor?

    1. A4
    2. B8
    3. C16
    4. D64
    Show answerHide answer

    C: 16

    \(4^2 = 16\).

  3. Two similar shapes have area ratio \(1 : 9\). What is the length ratio?

    1. A\(1 : 81\)
    2. B\(1 : 3\)
    3. C\(1 : 9\)
    4. D\(1 : 4.5\)
    Show answerHide answer

    B: \(1 : 3\)

    \(\sqrt{9} = 3\), so \(1 : 3\).

  4. A shape with area 5 cm² is enlarged by scale factor 2. What is the new area?

    1. A10 cm²
    2. B25 cm²
    3. C7 cm²
    4. D20 cm²
    Show answerHide answer

    D: 20 cm²

    \(5 \times 4 = 20\) cm².

  5. In triangle ABC with DE parallel to BC, \(AD = 3\), \(AB = 9\) and \(DE = 4\). What is \(BC\)?

    1. A7
    2. B9
    3. C12
    4. D27
    Show answerHide answer

    C: 12

    Scale factor 3, so \(BC = 12\).

  6. Two similar triangles have areas 20 cm² and 45 cm². What is the length scale factor from the smaller to the larger?

    1. A1.5
    2. B2.25
    3. C25
    4. D3
    Show answerHide answer

    A: 1.5

    \(\dfrac{45}{20} = 2.25\), and \(\sqrt{2.25} = 1.5\).

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